Section 6.3 Factored Form of a Quadratic Function
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Review: Determine the characteristics of a quadratic function given the graph.
Given the graph of a quadratic function, state the following:
(a) the vertex
(b) the equation of the axis of symmetry
(c) the coordinates of the y–intercept
(d) the number of x–intercepts
(e) the coordinates of the xintercepts
Example:
Note: When we state the coordinates of the xintercepts, the y–coordinate is always ____.
vertex yintercept axis of symmetryxintercept(s)
Section 6.3 Factored Form of a Quadratic Function
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Review:Determine the characteristics of a quadratic function given the equation.
Example:For the quadratic function y = x2 – 8x + 12 determine:
(a) the coordinates of the y – intercept
(b) the vertex
(c) the equation of the axis of symmetry
(d) the number of xintercepts
Section 6.3 Factored Form of a Quadratic Function
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Section 6.3: Factored Form of a Quadratic Function
make the connection between the factored form of a quadratic and the xintercepts of the graph
Example:
(i) Standard Form
(ii) Factored Form
Forms of a Quadratic Function
Factored Form
Standard Form
Example:
Factored Form
Standard Form
Section 6.3 Factored Form of a Quadratic Function
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Investigate how to algebraically attain x – intercepts from a
quadratic function in standard form y = ax2 + bx + c.
xintercepts from a Quadratic Function
Graph each of the quadratic functions:
https://www.desmos.com/
(a)
coordinates of vertex:____
axis of symmetry:_____
coordinates of y – intercept:_____
coordinates of x – intercepts:__________
What is the standard form of the quadratic function?
Section 6.3 Factored Form of a Quadratic Function
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(b)
coordinates of vertex:____
axis of symmetry:_____
coordinates of y – intercept:_____
coordinates of x – intercepts:__________
What is the standard form of the quadratic function?
Section 6.3 Factored Form of a Quadratic Function
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Using the results of the quadratic functions above and their respective graphs, answer each of the questions.
1. Which form of the quadratic function is easiest for determining the
x – intercepts without the graph?
2. What is the connection between the factored form of a quadratic function and the x – intercepts?
(i) Standard Form
(ii) Factored Form
Forms of a Quadratic Function
3. What is the value of the y – coordinate at the point where the graph
crosses the x – axis?
Example:
Section 6.3 Factored Form of a Quadratic Function
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Algebraically Determine the x – intercepts of a Quadratic Function in Factored Form
the xintercepts of the graph of a function is the same as the roots of the equation or the zeros of a function
Possible Number of xintercepts for a Quadratic Graph
Section 6.3 Factored Form of a Quadratic Function
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x – intercepts of a Quadratic Function
Examples:
Zero Product Property
If the product of two real numbers is zero ( a • b = 0)
then one or both must be zero.
In other words: a = 0 and b = 0
Section 6.3 Factored Form of a Quadratic Function
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Example 1: Given the quadratic function y = –(x + 2)(x – 4)
(a) determine the x – intercepts
(b) determine the axis of symmetry
(c) determine the coordinates of the vertex
(d) determine the y – intercept
(e) sketch the graph
(f) state the range
Section 6.3 Factored Form of a Quadratic Function
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Example 2: Given the quadratic function y = 2x( x + 4)
(a) determine the x – intercepts
(b) determine the axis of symmetry
(c) determine the coordinates of the vertex
(d) determine the y – intercept
(e) sketch the graph
(f) state the range
Section 6.3 Factored Form of a Quadratic Function
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Example 3: Given the quadratic function y = –3(x – 1)(x – 1)
(a) determine the x – intercepts
(b) determine the axis of symmetry
(c) determine the coordinates of the vertex
(d) determine the y – intercept
(e) sketch the graph
(f) state the range
PRACTICE: p. 346 – 347 #1, 2, 4ade, 5
Section 6.3 Factored Form of a Quadratic Function
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Determine the Equation of a Quadratic Function (Standard Form)
Example 1: Determine the function that defines this parabola.
Express the function in standard form.
write the equation in factored form and then expand to standard form
determine the xintercepts and the value.
Step 1: Using the xintercepts express the function in factored form
Step 2: Use the coordinates of another point to solve
for the value of
Step 3: Expand factored form
to produce standard form
Section 6.3 Factored Form of a Quadratic Function
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Example 2:
Determine the function that defines this parabola.
Express the function in standard form.
Section 6.3 Factored Form of a Quadratic Function
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Determining the Equation of a Quadratic Function Based on a Verbal Description.
(a) Determine the quadratic function that models the height of the missile over time.
(b) State the domain and range of the variables.
Domain:_________________
Range:__________________
PRACTICE: p. 348 – 349 #11ad, 13, 15, 17
Example:
A missile fired from ground level attains a height of 180 m at 2 seconds. The missile is in the air for 6 seconds
Section 6.3 Factored Form of a Quadratic Function
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Maximum/Minimum Word Problems
(A) Determining the maximum height given the quadratic function.
Example 1:
A boat in distress fires off a flare. The height of the
flare, h, in metres above the water, t seconds
after shooting, is modeled by the function
h(t) = –4.9t2 + 29.4t + 3.
Algebraically determine the maximum height attained by the flare.
to solve max/min problems you will have to determine the highest (or lowest) point , in other words, the vertex.
Types:
1) Quadratic Equation is Given
2) Create a Quadratic EquationArea
Revenue
Section 6.3 Factored Form of a Quadratic Function
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Example 2:
The path of a volleyball is given by where t is the
time in seconds and h is the height in meters. At what time does the ball reach its maximum height?
Please note we do not have to go further to substitute it back into the equation to find the height. This question asks just when it happens.
Be very careful to INTREPRET the question correctly!
Section 6.3 Factored Form of a Quadratic Function
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(B) Area Problems
(i) Open Field
Example 1:
A farmer is constructing a rectangular fence in an open field to contain
cows. There is 120 m of fencing.
(a) Write the quadratic function that models the region.
(b) Determine the maximum area of the enclosed region.
Section 6.3 Factored Form of a Quadratic Function
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Example 2:
A rectangular play enclosure for some dogs is to be made with 60 m of fencing using the kennel as one side of the enclosure as shown.
(a) Algebraically determine the quadratic function that models the area.
(i) Using a Physical Structure as One Side
(b) Use the function to determine the maximum area.
(c) State the domain and range of the variables in the function.
Section 6.3 Factored Form of a Quadratic Function
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Example 3:
A rectangular region, placed against the wall of a house, is divided into three regions of equal area using a total of 120 m of fencing as shown.
(a) Algebraically develop a quadratic function that models the area.
(b) Determine the maximum area of the pen.
(c) State the domain and range of the variables in the function.
Section 6.3 Factored Form of a Quadratic Function
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(B) Revenue Problems
Example 1:
Global Gym charges its adult members $50 monthly for a membership. The club has 600 adult members. Global Gym estimates that for each $5 increment in the monthly fee, it will lose 50 members.
(a) Determine the function that models Global Gym’s revenue.
(b) Determine the maximum revenue generated.
(c) Determine the monthly fee that will produce the greatest revenue.
Section 6.3 Factored Form of a Quadratic Function
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Example 2:
An orange grower has 400 crates of oranges ready for market and will have 20 more crates each day that shipment is delayed. The present price is $60 per crate however, for each day shipment is delayed, the price per crate decreases by $2.
(a) Determine the revenue function that models this function.
(b) Determine the maximum revenue that can be generated.
(c) Determine the price per crate that will produce the greatest revenue.
PRACTICE: p. 347, #8, 9 + Your Turn
Section 6.3 Factored Form of a Quadratic Function
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Your Turn
1. A dinner theatre show which sells out each night with 400 tickets currently
cost $10 per ticket. Proposed increases in ticket prices reveal that for each
$2 increase, 20 less people will attend. Write a quadratic function to model
the theatre’s revenue and determine the maximum revenue generated.
Section 6.3 Factored Form of a Quadratic Function
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2. An Airline company sells 500 tickets per flight at a cost of $100 per ticket.
Proposed increases in ticket prices reveal that for each $5 increase, 20 less
people will purchase tickets. Write a quadratic function to model the
Airline’s revenue per flight and use it to determine the maximum revenue
that can be generated per flight. What ticket price should the airline charge
to maximize revenue?
Section 6.3 Factored Form of a Quadratic Function
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3. A barn which contains different livestock will use 240 m of fencing to
construct three equal rectangular regions. There is no fencing along the
side of the barn so livestock can move in and out of the barn. The
quadratic function A(x) = –4x2 + 240x models the area of the pen where
A(x) represents the maximum area and x represents the width.
(a) Determine the maximum area of the pen.
(b) State the domain and range.
Section 6.3 Factored Form of a Quadratic Function
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4. A barn which contains different livestock will use 240 m of fencing to construct three equal rectangular regions. There is no fencing along the side of the barn so livestock can move in and out of the barn.
(a) Develop a quadratic function that
models the area of the pen.
(b) Determine the maximum area of
the pen and state the dimensions.
Section 6.3 Factored Form of a Quadratic Function
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5. A lifeguard marks off a rectangular swimming area at a beach with 200 m
of rope using the beach as one side. Determine the maximum area and the
dimensions of the swimming area?
Section 6.3 Factored Form of a Quadratic Function
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6. A farmer is going to construct a rectangular fence in an open field using
400m of fencing. Develop an appropriate quadratic function and use it to
determine the maximum enclosed area and the dimensions of the
rectangular region.
Section 6.3 Factored Form of a Quadratic Function
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7. A rectangular storage area for heavy equipment is to be constructed using
148 m of fencing and a building as one side.
Set up an appropriate equation and use it to determine the dimensions required to maximize the area enclosed.